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Licensed Unlicensed Requires Authentication Published by De Gruyter March 10, 2015

Uniform bounds of minimizers of non-smooth constrained functionals on maps spaces

Jurandir Ceccon and Marcos Montenegro

Abstract

We consider the functional Φ(u) = ∫Ω |∇u|2dx - ∫ΩG(u)dx constrained to the set EF = {uW01,2(Ω,ℝk) : ∫ΩF(u)dx = 1}, where Ω is a bounded open subset of ℝn and F,G : ℝk → ℝ are continuous functions satisfying certain homogeneity conditions. We investigate the L regularity of minimizers of Φ in EF. Moreover, we establish uniform L bounds for such minimizers as well as concentration results on Ω̅. In the latter case, we prove that, up to dilations and translations, minimizers behave in a certain sense like a special type of vector bubble. The central difficulty in this study is the fact that the minimizers of Φ do not have an Euler–Lagrange equation associated.

MSC: 35A15; 35J60

Funding source: Capes

Funding source: CNPq

Funding source: Fapemig

The authors are indebted to the referee for his (her) valuable suggestions and comments pointed out concerning this work.

Received: 2014-6-15
Revised: 2014-11-25
Accepted: 2015-2-26
Published Online: 2015-3-10
Published in Print: 2016-4-1

© 2016 by De Gruyter

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